Epsilondelta limit definition 1  Limits  Differential Calculus  Khan Academy  Summary and Q&A
TL;DR
This video explains the rigorous definition of a limit, which involves finding a range around a point where the function approaches a specific value.
Questions & Answers
Q: What does the limit of a function as x approaches a represent?
The limit represents the value that the function approaches as x gets closer to the given point a from both the positive and negative sides.
Q: How is the epsilondelta definition of limits different from the intuitive understanding of limits?
The epsilondelta definition provides a more mathematically rigorous explanation of limits by specifying a range around the point a, where the function will be within a given distance (epsilon) from the limit.
Q: Can you give an example of how the epsilondelta definition works?
Sure! Let's say we want to find the limit of a function as x approaches 2. We choose epsilon to be 0.5, and according to the definition, we can find a delta such that if x is within delta of 2, the function will be within 0.5 of the limit.
Q: What happens if x is equal to a in the epsilondelta definition?
The definition does not apply when x is equal to a because the function may be undefined at that point. The definition only guarantees the behavior of the function when x is within the specified range around a.
Summary & Key Takeaways

The video introduces the concept of limits and provides a visual representation of a function with a hole at a specific point.

It explains that the limit of a function as x approaches a is the value that the function approaches as x gets closer to a from both sides.

The video then introduces the epsilondelta definition of limits, which states that for any given distance from the limit point (epsilon), there exists a range around x (delta), where the function will be within the specified distance from the limit.